A065338 -id:A065338 - cp's OEIS frontend

A319690 Fully multiplicative with a(p^e) = (p mod 3)^e.

1, 2, 0, 4, 2, 0, 1, 8, 0, 4, 2, 0, 1, 2, 0, 16, 2, 0, 1, 8, 0, 4, 2, 0, 4, 2, 0, 4, 2, 0, 1, 32, 0, 4, 2, 0, 1, 2, 0, 16, 2, 0, 1, 8, 0, 4, 2, 0, 1, 8, 0, 4, 2, 0, 4, 8, 0, 4, 2, 0, 1, 2, 0, 64, 2, 0, 1, 8, 0, 4, 2, 0, 1, 2, 0, 4, 2, 0, 1, 32, 0, 4, 2, 0, 4, 2, 0, 16, 2, 0, 1, 8, 0, 4, 2, 0, 1, 2, 0, 16, 2, 0, 1, 8, 0, 4, 2, 0, 1, 8, 0, 16, 2, 0, 4, 8, 0, 4

Offset: 1

Antti Karttunen, Oct 04 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A065338, A319691.

A319690(n) = { my(f=factor(n)); prod(i=1, #f~, (f[i, 1]%3)^f[i, 2]); };

A072436 Remove prime factors of form 4*k+3.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 13, 2, 5, 16, 17, 2, 1, 20, 1, 2, 1, 8, 25, 26, 1, 4, 29, 10, 1, 32, 1, 34, 5, 4, 37, 2, 13, 40, 41, 2, 1, 4, 5, 2, 1, 16, 1, 50, 17, 52, 53, 2, 5, 8, 1, 58, 1, 20, 61, 2, 1, 64, 65, 2, 1, 68, 1, 10, 1, 8, 73, 74, 25, 4, 1, 26, 1, 80, 1, 82, 1, 4, 85, 2, 29

Offset: 1

Reinhard Zumkeller, Jun 17 2002

a(n) <= n; a(a(n)) = a(n); for all factors p^m of a(n): p=2 or p=4*k+1.

			a(90) = a(2*3*3*5) = a(2*(4*0+3)^2*(4*1+1)^1) = 2*1^2*5 = 10.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A072438, A072437, A002144, A002145, A065338.

Equals n / A097706(n).

a:= n-> mul(`if`(irem(i[1], 4)=3, 1, i[1]^i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 09 2014

a[n_] := n/Product[{p, e} = pe; If[Mod[p, 4] == 3, p^e, 1], {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, May 29 2019 *)

a(n) = my(f=factor(n)); for (k=1, #f~, if ((f[k,1] % 4) == 3, f[k,1]=1)); factorback(f); \\ Michel Marcus, May 08 2017

from sympy import factorint
from operator import mul
def a(n):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==3 else i**f[i] for i in f])# Indranil Ghosh, May 08 2017

Multiplicative with a(p) = (if p==3 (mod 4) then 1 else p).

A319984 Fully multiplicative with a(p^e) = prime(p mod 4)^e.

Original entry on oeis.org

1, 3, 5, 9, 2, 15, 5, 27, 25, 6, 5, 45, 2, 15, 10, 81, 2, 75, 5, 18, 25, 15, 5, 135, 4, 6, 125, 45, 2, 30, 5, 243, 25, 6, 10, 225, 2, 15, 10, 54, 2, 75, 5, 45, 50, 15, 5, 405, 25, 12, 10, 18, 2, 375, 10, 135, 25, 6, 5, 90, 2, 15, 125, 729, 4, 75, 5, 18, 25, 30, 5, 675, 2, 6, 20, 45, 25, 30, 5, 162, 625, 6, 5, 225, 4, 15, 10, 135, 2, 150, 10, 45, 25, 15, 10

Offset: 1

Antti Karttunen, Oct 06 2018

For all i, j:

A319714(i) = A319714(j) => a(i) = a(j) => A065338(i) = A065338(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. also A065338, A319714, A319985, A319986, A319987, A320114, A320115.

A319984(n) = { my(f=factor(n)); prod(i=1, #f~, (prime(f[i, 1]%4))^f[i, 2]); };

For all n, A003963(a(n)) = A065338(n).

A246269 a(1) = 1, a(p(k)) = p(k+1) mod 4 for k-th prime p(k) and a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

1, 3, 1, 9, 3, 3, 3, 27, 1, 9, 1, 9, 1, 9, 3, 81, 3, 3, 3, 27, 3, 3, 1, 27, 9, 3, 1, 27, 3, 9, 1, 243, 1, 9, 9, 9, 1, 9, 1, 81, 3, 9, 3, 9, 3, 3, 1, 81, 9, 27, 3, 9, 3, 3, 3, 81, 3, 9, 1, 27, 3, 3, 3, 729, 3, 3, 3, 27, 1, 27, 1, 27, 3, 3, 9, 27, 3, 3, 3, 243, 1, 9, 1, 27, 9, 9, 3, 27

Offset: 1

Antti Karttunen, Aug 21 2014

This is a fully multiplicative sequence. Only powers of 3 (A000244) occur as terms.

			For n = 10 = 2*5 = p_1 * p_3 we have a(n) = (p_{1+1} mod 4)*(p_{3+1} mod 4) = (p_2 mod 4) * (p_4 mod 4) = (3 mod 4)*(7 mod 4) = 3*3 = 9.

Antti Karttunen, Table of n, a(n) for n = 1..10080

Cf. A000244, A003961, A065338, A246270, A246272.

default(primelimit, 2^22)
A246269(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = (nextprime(f[i, 1]+1)%4)); factorback(f);
for(n=1, 10080, write("b246269.txt", n, " ", A246269(n)));

(define (A246269 n) (A065338 (A003961 n)))

a(n) = A065338(A003961(n)).

a(n) = A000244(A246270(n)).

A240370 Positive integers n such that every element in the ring of integers modulo n can be written as the sum of two squares modulo n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 122, 123, 125, 127, 129, 130, 131, 133, 134, 137, 138, 139, 141, 142, 143, 145, 146, 149, 150, 151, 154, 155, 157, 158, 159, 161, 163, 165, 166, 167, 169

Offset: 1

Charles R Greathouse IV, Apr 04 2014

nonn

Numbers n such that, if p^2 divides n for any prime p, then p = 1 mod 4.
Equivalently, squarefree numbers times A004613.
Thus, numbers k such that A065338(A057521(k)) = 1. - Antti Karttunen, Jun 21 2014
Different from A193304: terms 169, 289, 338, 507, 578, 841, 845, 867, ... are here but not in A193304. - Michel Marcus, Jun 20 2014
The asymptotic density of this sequence is 3/(8*K^2) = (3/4) * A243379 = 0.64208..., where K is the Landau-Ramanujan constant (A064533). - Amiram Eldar, Dec 19 2020

			In Z_7, 0^2 + 0^2 = 0, 1^2 + 0^2 = 1, 1^2 + 1^2 = 2, 3^2 + 1^2 = 3, 2^2 + 0^2 = 4, 2^2 + 1^2 = 5, 3^2 + 2^2 = 6. Therefore 7 is in the sequence.
In Z_8, there is no way to express 3 as a sum of two squares. Therefore 8 is not in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Joshua Harrington, Lenny Jones, and Alicia Lamarche, Representing Integers as the Sum of Two Squares in the Ring Z_n, arXiv:1404.0187 [math.NT], Apr 01 2014 and, J. Int. Seq. 17 (2014) # 14.7.4.

The subsequence A240109 is a version not allowing 0.
Different from A193304.
Complement of A053443. Subsequence of A192450.
Cf. A004613, A057521, A064533, A065338, A243379.

rad[n_] := Times @@ First /@ FactorInteger[n];
a57521[n_] := n/Denominator[n/rad[n]^2];
a65338[n_] := a65338[n] = If[n==1, 1, Mod[p = FactorInteger[n][[1, 1]], 4]* a65338[n/p]];
Select[Range[200], a65338[a57521[#]] == 1&] (* Jean-François Alcover, Sep 22 2018, after Antti Karttunen *)
Select[Range[200], AllTrue[FactorInteger[#], Mod[First[#1], 4] == 1 || Last[#1] == 1 &] &] (* Amiram Eldar, Dec 19 2020 *)

is(n)=my(f=factor(n)); for(i=1,#f~,if(f[i,2]>1 && f[i,1]%4>1, return(0))); 1

isok(n) = { if (n < 2, return (0)); if ((n % 4) == 0, return (0)); forprime(q = 2, n, if (((q % 4) == 3) && ((n % q) == 0) && ((n % q^2) == 0), return (0)); ); return (1); } \\ Michel Marcus, Jun 08 2014

;; With Antti Karttunen's IntSeq-library.
(define A240370 (MATCHING-POS 1 1 (lambda (k) (= 1 (A065338 (A057521 k))))))
;; Antti Karttunen, Jun 21 2014

A246349 Positions of records in A246272.

Original entry on oeis.org

1, 2, 6, 10, 30, 42, 210, 330, 462, 2310, 6090, 30030, 66990, 94710, 434910, 651630, 1292646, 1610070, 2478630, 2497110, 2916690, 13220130, 20930910, 52582530, 60690630

Offset: 1

Antti Karttunen, Aug 23 2014

nonn

All terms are squarefree. (See the comments in A246272).

From 2 onward they factorize as: 2, 2*3, 2*5, 2*3*5, 2*3*7, 2*3*5*7, 2*3*5*11, 2*3*7*11, 2*3*5*7*11, 2*3*5*7*29, 2*3*5*7*11*13, 2*3*5*7*11*29, 2*3*5*7*11*41, 2*3*5*7*19*109, 2*3*5*7*29*107, 2*3*17*19*23*29, 2*3*5*7*11*17*41, 2*3*5*7*11*29*37, 2*3*5*7*11*23*47, 2*3*5*7*17*19*43, 2*3*5*7*11*59*97, 2*3*5*7*11*13*17*41, 2*3*5*7*11*13*17*103, 2*3*5*7*11*13*43*47, ...

A246350 gives the corresponding record values.

Cf. A246272.

default(primelimit, 2^22)
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A065338(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = (f[i, 1]%4)); factorback(f);
A246272(n) = {my(i); i=0; while((A065338(n)!=1), i++; n = A003961(n)); i};
\\ Compute the b-files for both the positions of records (A246349) and their values (A246350) at the same time:
prevmax = -1; i = 0; for(n=1, 60690630, if((k=A246272(n)) > prevmax, prevmax = k; i++; write("b246349.txt", i, " ", n); write("b246350.txt", i, " ", k)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246349 (RECORD-POS 1 1 A246272))

A278265 a(n) = A278509(A276573(n)).

Original entry on oeis.org

3, 3, 1, 3, 3, 1, 9, 9, 3, 27, 3, 1, 3, 3, 1, 3, 9, 3, 3, 1, 3, 3, 27, 1, 3, 3, 9, 3, 3, 1, 3, 1, 3, 9, 9, 3, 27, 3, 9, 27, 3, 3, 9, 3, 3, 27, 1, 3, 3, 1, 3, 3, 9, 27, 1, 3, 3, 3, 1, 81, 9, 9, 27, 1, 3, 3, 3, 9, 81, 3, 3, 27, 1, 3, 27, 9, 9, 27, 3, 1, 3, 3, 27, 1, 3, 3, 243, 9, 3, 3, 3, 1, 3, 9, 9, 3, 27, 1, 3, 1, 3, 3, 3, 9, 3, 27, 1, 1, 9, 3, 9, 9, 3, 27, 3

Offset: 1

Antti Karttunen, Nov 28 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10028

Cf. A000265, A065338, A276573, A276575, A278498, A278509.

(define (A278265 n) (A000265 (A278498 n)))

a(n) = A278509(A276573(n)) = A065338(A000265(A276573(n))).

a(n) = A000265(A278498(n)).

A278509 a(n) = 3^{number of primes congruent to 3 modulo 4 dividing n (with multiplicity)}.

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 3, 1, 9, 1, 3, 3, 1, 3, 3, 1, 1, 9, 3, 1, 9, 3, 3, 3, 1, 1, 27, 3, 1, 3, 3, 1, 9, 1, 3, 9, 1, 3, 3, 1, 1, 9, 3, 3, 9, 3, 3, 3, 9, 1, 3, 1, 1, 27, 3, 3, 9, 1, 3, 3, 1, 3, 27, 1, 1, 9, 3, 1, 9, 3, 3, 9, 1, 1, 3, 3, 9, 3, 3, 1, 81, 1, 3, 9, 1, 3, 3, 3, 1, 9, 3, 3, 9, 3, 3, 3, 1, 9, 27, 1, 1, 3, 3, 1, 9, 1, 3, 27, 1, 3, 3, 3, 1, 9, 3, 1, 9, 3, 3, 3

Offset: 1

Antti Karttunen, Nov 28 2016

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000244, A000265, A065338, A065339.

Cf. also A278265.

f[p_, e_] := Mod[p, 4]^e; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 13 2023 *)

(define (A278509 n) (A065338 (A000265 n)))

Fully multiplicative with a(p^e) = 1 if p = 2, (p mod 4)^e if p > 2.
a(n) = A065338(A000265(n)) = A000265(A065338(n)).
a(n) = A000244(A065339(n)) = 3^A065339(n).

A339875 Intersection of A137409 and A339870: Composite numbers k of the form 4u+1 having more than one prime factor of type 4u+3, and for which the odd part of phi(k) divides k-1.

Original entry on oeis.org

561, 6601, 8481, 17733, 23001, 30889, 54741, 62745, 88561, 106141, 319965, 359601, 449065, 534061, 609301, 949785, 1357621, 2162721, 2288661, 2615977, 3284281, 4005001, 4698001, 4830805, 5381265, 6313681, 6594721, 6840001, 8093701, 11782005, 11921001, 14665105, 14892153, 15217741, 16577785, 19683001, 20154061, 20441701

Offset: 1

Antti Karttunen, Dec 26 2020

nonn

Composite numbers k of the form 4u+1 for which the odd part of phi(k) divides k-1 and for which A065338(k) > 1.

All terms k are squarefree and the 3-adic valuation of A065338(k) is a nonzero even number.

Intersection of A137409 and A339870.

Cf. A000010, A000265, A053575, A065338.

A065338(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = (f[i, 1]%4)); factorback(f); };
isA137409(n) = ((1==n)||(A065338(n)>1)); \\ Antti Karttunen, Dec 26 2020
A000265(n) = (n>>valuation(n, 2));
isA339870(n) = ((n>1)&&!isprime(n)&&(1==(n%4))&&!((n-1)%A000265(eulerphi(n))));
isA339875(n) = (isA137409(n)&&isA339870(n));

cp's OEIS Frontend

A319690 Fully multiplicative with a(p^e) = (p mod 3)^e.

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A072436 Remove prime factors of form 4*k+3.

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A319984 Fully multiplicative with a(p^e) = prime(p mod 4)^e.

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A246269 a(1) = 1, a(p(k)) = p(k+1) mod 4 for k-th prime p(k) and a(u * v) = a(u) * a(v) for u, v > 0.

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A240370 Positive integers n such that every element in the ring of integers modulo n can be written as the sum of two squares modulo n.

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Mathematica

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A246349 Positions of records in A246272.

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A278265 a(n) = A278509(A276573(n)).

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A278509 a(n) = 3^{number of primes congruent to 3 modulo 4 dividing n (with multiplicity)}.

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